A consistency proof for some restrictions of Tait's reflection principles

In 5, Tait identifies a set of reflection principles called \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Gamma ^{(2)}_{n}$\end{document} ‐reflection principles which Peter Koellner has shown to be consistent relative to the existence of κ(ω), the first ω‐Erdős cardinal 1. Tait also defines a set of reflection principles called \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Gamma ^{(m)}_{n}$\end{document} ‐reflection principles; however, Koellner has shown that these are inconsistent when m > 2, but identifies restricted versions of them which he proves consistent relative to κ(ω) 2. In this paper, we introduce a new large‐cardinal property, the α‐reflective cardinals. Their definition is motivated by Tait's remarks on parameters of third or higher order in reflection principles. We prove that if κ is ℶ κ + α + 1 ‐supercompact and 0 < α < κ then κ is α‐reflective. Furthermore we show that α‐reflective cardinals relativize to L, and that if κ(ω) exists then the set of cardinals λ < κ(ω), such that λ is α‐reflective for all α such that 0 < α < λ, is a stationary subset of κ(ω). We show that an ω‐reflective cardinal satisfies some restricted versions of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Gamma ^{(m)}_{n}$\end{document} ‐reflection, as well as all the reflection properties proved consistent in 2.